Papers. On the resistance law in a shallow sea. Abstract

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Papers On the resistance law in a shallow sea Bottom stress Shallow sea R afał L e w a n d o w ic z Institute of Meteorology and Water Management Gdynia Manuscript received June 27, 1984, in final

1 Papers On the resistance law in a shallow sea Bottom stress Shallow sea R afał L e w a n d o w ic z Institute of Meteorology and Water Management Gdynia Manuscript received June 27, 1984, in final form December 21, Abstract Assuming a two-layer structure of turbulence in a homogeneous shallow sea, expressions describing the geostrophic coefficient of bottom friction «,/G and the angle between the vector of geostrophic current and the vector of tangential stress at the bottom were obtained as functions of non-dimensional parameters. The generally accepted method of determining the effect of bottom friction in a shallow sea is based on the use of the square law of friction, in which the coefficient of bottom friction is a constant quantity or is defined by flow parameters [4]. Besides, it is known that in a shallow sea with neutral stratification, the vertical distribution of velocity and the turbulence characteristics determining the magnitude and direction of bottom stress are functions of normalized with respect to sea water density components of wind tangential stress at the free surface (t //?, t //?), velocity of the geostrophic current (G), depth (H), Coriolis parameter (/), and the sea bottom roughness parameter (z0). According to the 7r-theorem of the similarity theory, four nondimensional combinations may be formed out of the six above-mentioned parameters: t ax\p G2, t y/p G2, G/fz0, H/z0. It may be expected that all turbulence characteristics in a shallow sea with neutral stratification, including the magnitude and direction of the bottom stress vector, will be related to the external defining parameters only by the above-metioned non-dimensional combinations. The aim of this paper is to obtain expressions describing the geostrophic coefficient of bottom friction G and the angle a between the vector of geostrophic current and the direction of tangential stress at the bottom (veering angle) as a function of t '/p G2, Tyjp G2, G/fzo, Hlz0. We take the traditional description of velocity and structure of turbulence in a shallow homogeneous sea as our starting point. We thus assume that in the thin layer close to the bottom the vertical distribution of velocity is determined by the logarithmic law (correspondingly, the vertical eddy-viscosity coefficient is a linear

function of the vertical coordinate), and at the interface betweçn the near-bottom logarithmic layer and the layer above it, the coefficient of turbulent viscosity and the momentum flux are

2 function of the vertical coordinate), and at the interface betweçn the near-bottom logarithmic layer and the layer above it, the coefficient of turbulent viscosity and the momentum flux are continuous. In this case, the equation of motion and the boundary conditions describing the vertical velocity distribution above the logarithmic layer take the following form: 1 d du v G sin a-i k = 0, / oz 1 ô 8v u G cos a k = 0, / du dv k =Tx/p; k = T ip at z = li, cu Q dv n k =Tx/p; k- =xyip at z = h, (D (2) u, v current velocity components in the direction of x and y axes respectively (x axis is directed along the vector of bottom friction), 1 lfdp\2 (dp\2 G geostrophic current velocity modulus (G = /I I +( ), where, p water density,/ Coriolis parameter, andp hydrostatic pressure), k eddy-viscosity coefficient, h thickness of the logarithmic bottom layer, z axis directed vertically upwards. The origin of the coordinate system is situated at the sea bottom. Let us assume that the eddy-viscosity coefficient k in the layer h ^z^h is constant and, according to the continuity condition at a depth z h, equals: k-ku^h, w* = frictional velocity, /c=von Karman constant. We now use the above-mentioned logarithmic velocity profile in the near-bottom layer and the continuity condition of velocity at a level z = h. As a result we have: (3) /l I/* - In------G cosa = Fu(h) k z0 K G sin a = Fv(h) K (4) ; -if M. (5, K J I = A K f z = h

3 We determine the thickness h of the logarithmic layer by assuming that is proportional to w* //, ie: h = c1 f ct = numerical constant. (6) Comparing expressions (4) and (5), and taking into account (6), after certain transformations, we obtain: lnr0= B -ln (u J G )+ *' -2- A 2, M {ujg y A w* sina= , k G (7) R 0 Rossby number (i?0 = G//zo), k d z^lp A - F M t f. / «. 3- (8) K d x j p B=Fu(h )-\n c1 = /«* d z z=h -ln ci Equations (7) correspond to the expressions describing the resistance law in the boundary planetary layer of the atmosphere (cf [1, 3, 4]). However, in our case (in a shallow sea), unlike in the neutrally stratified planetary layer of the atmosphere, A and B are certain functions of non-dimensional parameters rupg2, xÿ/p G2, R0, HJz0. Let us find the form of these functions. It appears from formulas (8) that A and B may be considered to be known when the distribution rxjp, xyjp in the layer h ^ z ^ H is known, and also the values of fcjp 8zylp at the level z=h. In order to determine t xjp, xy\p as functions of z, we Oz differentiate equations (1) with respect to z and, taking into account the following expressions: du dv t / p = t jc/ p + ÎT / p = A: h/a:, (9) as a result we obtain equation: ^ 2t/p.. /

$We solve the above equation with the following boundary conditions: T/p = Ta/P T/p = z /p at Z~H,\ at z h.$

4 We solve the above equation with the following boundary conditions: T/p = Ta/P T/p = z /p at Z~H,\ at z h. j (11) The solution of problem (10), (11) has the following form: x t 7p sh (1 + i) p (h - z) - T lp sh (1 + i) P(H-z) sh(l + i)0(h-ii) where [i=\!f/2k. Next we obtain: fc/p (l + 0 ^ l * a!p - T /p ch (1 + i) p (H -/i)]. (12) sh(l + i))?(h -ft)l We now take into consideration that for a shallow sea, inequalities h<<h and fih< 1 are true. Thus, replacing the hyperbolic functions in (12) by their series with respect to argument p H and limiting ourselves to the first terms, we obtain: dr/p = ~ i «l p - u l ) + izay/p2, (13) h H where the frictional velocity is determined by h* = V t /p. Separating in (13) the real part from the imaginary one and replacing them in (8), we have: k[r"/pg2r0(h/z0)~1 (ujg)~2 - R0(H/zoy x], (jr B = K(uJG) - 1 -taylpg2r0(hlz0r 1 - I n Cl. (14) Equations (7) and (14) determine the resistance law in a shallow sea. Of course, this law holds only for a sea whose depth is limited by the condition ( H< 1, which after substituting expressions (3), (6) may be presented as: H < ~ \! 2 c t k. (15) It should be emphasized that depth H cannot be too small because in such a case the effect of the Coriolis force vanishes and equations (1) describe the vertical velocity profile incorrectly. Let us consider a special case of the resistance law (7) and (14), when there is no wind above the free surface of the sea. In this case, the defining parameters do not include the terms r /p, ty/p and the expressions for functions A and B take the form: A = ku-^r0(hiz0)~1; (j B InCi.

5 Fig. 1. Geostrophic coefficient of bottom friction (k*/g) as a function of Rossby number (R0) for different values of parameter if /r 0; c1 = H /zo = l-103, 2 - H / z o=3-103; 3 - H l z 0 = MO4; 4 - H ->oo Fig. 2. Rotation angle a as a function of Rossby number (i?0) for different values of parameter i3/z0; Ci = Explanations see Figure 1

$Substituting (16) in (7) we obtain the following form of the resistance law: ln R0=B - l n (ujg)+ L - V/c2 - A '\u J G f («*/G) yl' A', sina=/------(ujg) = (w*/g), K A = kr0(h/z0).$

6 Substituting (16) in (7) we obtain the following form of the resistance law: ln R0=B - l n (ujg)+ L - V/c2 - A '\u J G f («*/G) yl' A', sina=/------(ujg) = (w*/g), K A = kr0(h/z0).. (18) The geostrophic coefficient of bottom friction u^/g and the angle between the bottom stress vector and the geostrophic current direction a versus the Rossby number and versus the non-dimensional depth H/z0 are shown in Figures 1 and 2. In the case of a deep sea, the defining parameters should not include i ax\p, t Ip, and H. Increasing the depth to infinity, we obtain (with the help of (12)): ôz i»5; e-f z = h ÇZ where, as before, /?= \ ///2Ar. Substituting (19) in (8) we have; /1 = Vk/2c1; jb= \JkI2c1 Incj. (20) Expressions (7) and (20) define the resistance law in the case of a deep sea. Thus, similarly to the planetary layer of the atmosphere with neutral stratification, A and B are universal constants. The dependence of the geostrophic coefficient of friction m* G and angle a on the Rossby number, corresponding to the case of a deep sea is presented in Figures 1 and 2 by curves 4. One should point out to the two peculiar features of the curves presented in Figure 1 and 2. Firstly, a stronger dependence of u^/g on R0 in a shallow sea, and secondly the reverse relationship between a and R0 in a shallow and deep sea. It only results from the fact that A is a constant value in a deep sea while for a shallow sea it is a function of the Rossby number. Let us also note that in the range of R0 values considered here the geostrophic coefficient of bottom friction u^/g changes by two orders of magnitude, and the veering angle by several tens of degrees. This means that inaccuracies in the real values of the geostrophic bottom friction coefficient w*/g and angle a can lead to significant errors in determining the bottom stress vector direction. It is similar when one applies the empirical square law of bottom friction because the bottom friction coefficient in this law is uniquely related to w* IG and a. (17) Acknowledgments The author is most grateful to Prof. B. A. Kagan of the Leningrad Branch of the Institute of Oceanology, the Academy of Sciences of the USSR, for critical comments and assistance while this work was in progress.

7 References» 1. B ro w n R. A., 1982, On Two-Layer Models and the Similarity Functions for the PBL, Boundary- Layer Meteorology, Vol V o lc in g e r N. E., P ia s k o v s k ii R. V., 1968, Osnovnyie okieanologicheskiie zadachi tieorii mielkoi vody, Gidrometeoizdat, Leningrad. 3. Z ilitin k ie v ic h S. S., 1970, Dinamika pogranichnogo sloia atmosfiery, Gidrometeoizdat, Leningrad. 4. Z ilitin k ie v ic h S. S., C h a lik o v D. V., 1968, O zakonakh soprotivlienia i obmiena tieplom i vlagoi pri vzaimodieistvi atmosfiery z podstielaiushchei povierkhnostiu, Fiz. Atm. Okieana, Vol. IV, 7.

THE EFFECT OF STRATIFICATION ON THE ROUGHNESS LENGTH AN DISPLACEMENT HEIGHT

THE EFFECT OF STRATIFICATION ON THE ROUGHNESS LENGTH AN DISPLACEMENT HEIGHT S. S. Zilitinkevich 1,2,3, I. Mammarella 1,2, A. Baklanov 4, and S. M. Joffre 2 1. Atmospheric Sciences,, Finland 2. Finnish